The rear A/C is a completely self-contained unit powered by an electric motor...it may have it's own expansion valve but it will NOT affect the front. Now the heat in the rear does steal water from the front, so if you ever have coolant disappearing it's the plastic "T" that feeds the rear...common to have them crack.

As for your low idle, very common problem after the battery dies or is disconnected. The computer has to re-learn EVERYTHING, even how to wipe it's ass again! Look for the "re-training" procedure on here in past posts/tech stuff.

Hey, at least it's not your compressor...it's $750!